Objectives: (1) A real number is a limit of a sequence of �nite decimal expressions. (2) Two real numbers that are not equal di_er by some positive distance and so must have di_erent decimal expansions. (3) The modulus |�| allows arguments about the distance between numbers to be written precisely. (4) The modulus and quanti�ers allow us to precisely de�ne what it means for a sequence of numbers to converge. (5) An increasing bounded sequence converges.